"I can confirm [Bayern sporting director] Hasan Salihamidzic and I were in Barcelona last Wednesday. And we reached an agreement with both the club and the player over a transfer," Rummenigge said.
"The player is expected here in Munich in the next days, Sunday or Monday at the latest. He'll do his medical and then sign his contract."
Earlier, ESPN's Samuel Marsden reported talks were at an "advanced" stage between the Bundesliga giants and Coutinho's parent club:
Coutinho has struggled to establish himself at the Camp Nou, while Bayern are still busy rebuilding after losing the influence of wingers Arjen Robben and Franck Ribery, as well as the flair of No. 10 James Rodriguez, earlier this summer.
Coutinho's standing at Barca is summed up by his omission from the match squad for Friday's La Liga opener away to Athletic Bilbao.
It's important there are places in the starting XI for big-money summer imports Antoine Griezmann and Frenkie de Jong. Both play in positions where Coutinho has found it difficult to exert his influence, with Griezmann a prolific source of goals from out wide, while De Jong is a progressive midfielder who bosses central areas.
Coutinho's ability to play in the middle or on the flanks would be a major asset for Bayern. Die Roten said goodbye to Robben and Ribery but have already taken steps to fill the void by adding Ivan Perisic on loan.
The arrival of the Croatia international is yet another example of Bayern's terrific use of the loan market in recent years.
Rodriguez spent the last two seasons in Germany while still on the books at Real Madrid. Replacing the Colombian's vision and technique may prove more difficult than offsetting the departures of Robben and Ribery.
Coutinho could fill the void, provided he can get back to his pre-Barcelona best. He was exceptional for Liverpool before joining Barca in 2018, combining skill with pace, ingenuity and shooting power.
Even in a down season, Coutinho helped himself to 11 goals and five assists in all competitions for the Blaugrana during the 2018/19 campaign. Yet despite the solid numbers, the 27-year-old has rarely seemed like a natural fit as a No. 10 in a Barca team geared around the creative instincts of roaming forward Lionel Messi.
Being in Messi's shadow doesn't mean Coutinho isn't ready for a top European club, though. He repaired his reputation with some quality displays to help Brazil win the 2019 Copa America this summer:
Snapping up a schemer this gifted and versatile on favourable loan terms would be a coup for a Bayern team needing more magic from the heart of midfield.